A number is chosen at random from 10 to 30 both inclusive. What is the probability that the number is divisible by 3?

1. Analyze the question

We need to find the probability that a number chosen at random from the set of integers $\{10, 11, 12, \dots, 30\}$ is divisible by 3.

2. Explain the concept

Probability is defined as:

• Total outcomes: Since the range is "inclusive," we include both 10 and 30. To find the total count of numbers in a range $[a, b]$, we use the formula: $(b - a) + 1$.
• Favorable outcomes: These are the numbers within that range that are multiples of 3.

3. Show the solution

Step 1: Determine the total number of outcomes. The range is from 10 to 30 inclusive. Total numbers = $(30 - 10) + 1 = 20 + 1 = 21$.

Step 2: List the numbers divisible by 3 in this range. We look for multiples of 3 starting from 10: 12, 15, 18, 21, 24, 27, 30.

Step 3: Count the favorable outcomes. Counting the list above: 1 (12), 2 (15), 3 (18), 4 (21), 5 (24), 6 (27), 7 (30). There are 7 favorable outcomes.

Step 4: Calculate the probability.

Simplifying the fraction by dividing both numerator and denominator by 7:

4. Identify the answer

Comparing our result to the options provided, the correct answer is C.

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5. Exam Tip

When a question says "inclusive," always remember to add 1 to the difference of the numbers. If you had calculated $30 - 10 = 20$, you would have missed one number, leading to an incorrect probability of $7/20$. Always write out the sequence if the range is small to avoid silly mistakes!